For several decades, there have been a varying number of books on Calculus following the classic line of mathematical thought, where Mathematics is taught for everybody by means of rigorous definitions, theorems, and carefully detailed and extensive demonstrations. For mathematical education into the XXI Century the students need to achieve ability in handling of present mathematical tools and concepts from the beginning of their courses. These needs can be achieved today by means of a paradigmatic change in the focus of mathematics teaching: to learn to develop ideas and to experiment and test those ideas in such way that students can verify their own inferences. In this paper we report an educational research in teaching and learning functions models according to a new paradigm in hands-on experimental mathematics, with applications in the real world, i.e. sciences and engineering by using Computer Algebra Systems. The study of functions is presented, focused into the framing of Exploratory Learning Systems, where students learn by means of the action and their participation in it. It is designed for teachers working together with students in a computer laboratory like hands-on workshops-type activities on other sciences. In this way students have a more “alive”, “realistic” and “accessible” touch in Calculus.

For several decades, there have been a varying number of books on Calculus following the classic line of mathematical thought, where Mathematics is taught for everybody by means of rigorous definitions, theorems, and carefully detailed and extensive demonstrations. For mathematical education into the XXI Century the students need to achieve ability in handling of present mathematical tools and concepts from the beginning of their courses. These needs can be achieved today by means of a paradigmatic change in the focus of mathematics teaching: to learn to develop ideas and to experiment and test those ideas in such way that students can verify their own inferences. In this paper we report an educational research in teaching and learning functions models according to a new paradigm in hands-on experimental mathematics, with applications in the real world, i.e. sciences and engineering by using Computer Algebra Systems. The study of functions is presented, focused into the framing of Exploratory Learning Systems, where students learn by means of the action and their participation in it. It is designed for teachers working together with students in a computer laboratory like hands-on workshops-type activities on other sciences. In this way students have a more “alive”, “realistic” and “accessible” touch in Calculus.